If a + b = 8, ab = 10, then the value of a 3+ b 3is:
272
Understanding the Algebraic Problem
This question asks us to find the value of \(a^3 + b^3\) given the sum \((a + b)\) and the product \((ab)\) of two numbers, \(a\) and \(b\).
We are provided with the following information:
- The sum of \(a\) and \(b\), which is \(a + b = 8\).
- The product of \(a\) and \(b\), which is \(ab = 10\).
Our goal is to calculate the value of \(a^3 + b^3\).
Key Formulas for Sum of Cubes
To solve this, we need to use algebraic identities. The formula for the sum of cubes, \(a^3 + b^3\), is particularly useful here. There are a couple of ways to express it, but one common form that relates to the sum \((a+b)\) and product \((ab)\) is:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Notice that this formula requires \(a+b\), \(ab\), and \(a^2 + b^2\). We are given \(a+b\) and \(ab\), so we first need to find the value of \(a^2 + b^2\).
Calculating \(a^2 + b^2\)
We can find \(a^2 + b^2\) using another common algebraic identity:
\((a + b)^2 = a^2 + 2ab + b^2\)
We can rearrange this identity to isolate \(a^2 + b^2\):
\(a^2 + b^2 = (a + b)^2 - 2ab\)
Now, we can substitute the given values \(a + b = 8\) and \(ab = 10\) into this rearranged formula:
\(a^2 + b^2 = (8)^2 - 2(10)\)
\(a^2 + b^2 = 64 - 20\)
\(a^2 + b^2 = 44\)
So, the value of \(a^2 + b^2\) is 44.
Finding the Value of \(a^3 + b^3\)
Now that we have \(a + b = 8\), \(ab = 10\), and \(a^2 + b^2 = 44\), we can use the sum of cubes formula:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Substitute the known values into the formula:
\(a^3 + b^3 = (8)(44 - 10)\)
First, calculate the value inside the second parenthesis:
\(44 - 10 = 34\)
Now, multiply this result by \((a + b)\), which is 8:
\(a^3 + b^3 = (8)(34)\)
\(a^3 + b^3 = 272\)
Therefore, the value of \(a^3 + b^3\) is 272.
Step-by-Step Calculation Summary
Here is a summary of the steps:
- Identify the given values: \(a + b = 8\) and \(ab = 10\).
- Recall the formula for the sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
- Recognize the need for \(a^2 + b^2\) and use the identity \((a + b)^2 = a^2 + 2ab + b^2\) to find it: \(a^2 + b^2 = (a + b)^2 - 2ab\).
- Substitute the given values into the formula for \(a^2 + b^2\): \(a^2 + b^2 = (8)^2 - 2(10) = 64 - 20 = 44\).
- Substitute the values for \(a + b\), \(ab\), and \(a^2 + b^2\) into the sum of cubes formula: \(a^3 + b^3 = (8)(44 - 10)\).
- Perform the final calculation: \(a^3 + b^3 = (8)(34) = 272\).
Algebraic Identity Revision Table
| Identity | Formula |
|---|---|
| Square of Sum | \((x + y)^2 = x^2 + 2xy + y^2\) |
| Square of Difference | \((x - y)^2 = x^2 - 2xy + y^2\) |
| Difference of Squares | \(x^2 - y^2 = (x + y)(x - y)\) |
| Sum of Cubes | \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\) |
| Difference of Cubes | \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\) |
Additional Algebraic Information
Algebraic identities are equations that are true for all possible values of the variables involved. They are fundamental tools in algebra for simplifying expressions, factoring polynomials, and solving equations.
The identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) can also be derived from the expansion of \((a+b)^3\). The expansion is \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\). We can rearrange this to get \(a^3 + b^3 = (a+b)^3 - 3a^2b - 3ab^2\). Factoring out \(3ab(a+b)\) gives \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\). Then, factoring out \((a+b)\) gives \(a^3 + b^3 = (a+b)((a+b)^2 - 3ab)\). Expanding \((a+b)^2\) gives \(a^3 + b^3 = (a+b)(a^2 + 2ab + b^2 - 3ab)\), which simplifies to \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). This shows the consistency of the formulas.
Knowing these identities helps solve problems efficiently, especially in exams.
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